Atomistic Theory of Ostwald Ripening and Disintegration of Supported Metal Particles under Reaction Conditions

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1 pubs.acs.org/jacs Atomistic Theory o Ostwal Ripening an Disintegration o Supporte Metal Particles uner Reaction Conitions Runhai Ouyang, Jin-Xun Liu, an Wei-Xue Li* State Key Laboratory o Catalysis, Dalian Institute o Chemical Physics, Chinese Acaemy o Sciences, Dalian , China *S Supporting Inormation ABSTRACT: Unerstaning Ostwal ripening an integration o supporte tal particles uner operating conitions has been o central importance in the stuy o sintering an persion o heterogeneous catalysts or long-term inustrial implentation. To achieve a quantitative escription o these complicate processes, an atomistic an generic theory taking into account the reaction environnt, particle size an morphology, an tal support interaction is evelope. It inclues (1) energetics o supporte tal particles, (2) ormation o monors (both the tal aatoms an tal reactant complexes) on supports, an (3) corresponing sintering rate equations an total activation energies, in the presence o reactants at arbitrary temperature an pressure. The thermoynamic criteria or the reactant assiste Ostwal ripening an inuce integration are ormulate, an the inluence o reactants on sintering kinetics an repersion are mappe out. Most energetics an kinetics barriers in the theory can be obtaine conveniently by irst-principles theory calculations. This allows or the rapi exploration o sintering an integration o supporte tal particles in huge phase space o structures an compositions uner various reaction environnts. General strategies o suppressing the sintering o the supporte tal particles an acilitating the repersions o the low surace area catalysts are propose. The theory is applie to TiO 2 (110) supporte Rh particles in the presence o carbon monoxie, an reprouces well the broa temperature, pressure, an particle size range over which the sintering an repersion occurre in such experints. The result also highlights the importance o the tal carbonyl complexes as monors or Ostwal ripening an integration o supporte tal catalysts in the presence o. 1. INTRODUCTION Transition tals have been use to catalyze a wie range o chemical reactions in heterogeneous catalysis, which plays an important role in energy conversion, chemicals prouction, an environntal protection. To be more accessible to reactants, transition tal catalysts are usually perse on a high surace area support, an corresponing size alls typically in the range o nanoters. 1 Although perse tal particles expose a large number o low coorination sites which coul act as the active sites an greatly enhance the catalytic activity, 2 6 a high ratio o these low coorination sites estabilizes the perse tal particles in the anti. Thus, the tal particles ten to agglorate an/or sinter, either by coalescence o smaller particles or by Ostwal ripening or the growth o a larger particle at the expense o a smaller one In the en, the overall activity o the tal particles ecreases with ti an eventually eactivates ue to the loss o the active surace area. To prevent the sintering, the proper supports shoul be selecte to stabilize the tal particles by ans o the tal support interaction (MSI), but so ar, its utilization is achieve mainly by trial an error. 12,13 To increase the lieti o inustrial catalysts, it is important to know how to suppress the particle sintering rate an how to reperse the eactivate catalysts ue to the sintering. A unantal unerstaning o the sintering chanism an kinetics at the microscopic level woul be highly valuable to provie insight into controlling these processes. The stuy o sintering is urther complicate by consiering catalytic reactions usually operate at elevate temperatures an pressures The presence o reactants coul aect an/ or inuce ramatically the sintering, ruption, an persion o supporte tal particles, as well as the crystalline suraces For instance, it was oun that, uner elevate carbon monoxie () partial pressures, supporte Rh particles were reaily integrate to the mononuclear Rhcarbonyl complexes At higher temperature, the Rh complexes ecompose, an the Rh aatoms release starte to agglorate an orm larger tal particles. Similarly, reactant-assiste ripening an integration ha also been oun when supporte tal particles were expose to oxygen, an the reason was attribute to the ormation o volatile oxygen tal complexes. It has also been suggeste that reactants coul change the wetting behavior o tal particles, causing them to sprea out on supports when the asorbate tal bon energy excees the ierence in energy between the tal tal an tal support boning. 48,49 Moreover, the strong interaction between asorbate an tal Receive: September 2, 2012 Publishe: December 31, Arican Chemical Society 1760 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

2 Journal o the Arican Chemical Society particle coul weaken the tal tal bon, 50 which woul acilitate the etachnt o the tal aatoms rom small particles, an eventually promote sintering an integration. Despite extensive stuies so ar, a clear chanistic unerstaning an a quantitative escription o reactants on the sintering an integration o supporte tal particles remains missing. Sintering kinetics o supporte tal particles an the elentary steps involve was pioneere by Wynblatt an Gjostein (WG). 7,8 Although sintering in the presence o an oxygen environnt an the ormation o the oxygen tal complexes as the transient monors was stuie in this work, it is unclear yet when the tal reactant complexes rather than the tal aatoms as ominant monors will orm uner the reaction conitions. In particular, how will the ormation o the tal reactant complexes epen on the reaction conitions, the tal particle size an shape, an the MSI? How will the tal reactant complexe ormation aect the ripening kinetics an the unerlying chanism? The ormation o the tal reactant complexes may also be involve eeply in the reactant inuce integration, a act that has been use wiely to reperse the low surace area tal catalysts ue to the sintering. 27,51 55 It is thereore important to entangle the role o the tal reactant complexes in the reactant assiste ripening an inuce integration o supporte tal particles. Here, we will ocus on Ostwal ripening using a surace iusion moel, or which the elentary steps typically inclue etachnt o the tal atoms rom smaller particles to orm monors, iusion o monors on supports, an attachnt towar larger particles. A major improvent o the WG theory ollowing the Ostwal ripening moel was obtaine by Campbell an co-workers by incorporating size-epenent surace energy in their moel an using an exponential unction in the ormulation o the ripening rate instea o a irst-orer approximation o the associate Talyor series. 11,56 It was oun that surace energy an morphology o supporte tal particles are sensitive to the reaction environnts, a act that coul aect the ripening process. 60 However, a theory o sintering an integration accounting or the inluence o reactant asorption on surace energy an morphology o the supporte tal particles is not available yet. To aress these questions, an atomistic theory o Ostwal ripening an integration o supporte tal particles in the presence o reactants was evelope an is irst presente in section 2. We propose that the strong boning between the reactant an tal aatom on supports is essential or the ormation o tal reactant complexes. Surace energy an chemical potential o supporte tal particles with asorption o reactants are erive, an the thermoynamic variables escribing the asorption o reactants on aatoms an ormation o the tal reactant complexes on support are eine. The criteria or reactant assiste Ostwal ripening an inuce integration are ormulate, an corresponing rate equations are erive. The theory is applie to TiO 2 (110) supporte Rh particles uner in Section 3, an most paraters require are calculate by irst-principles theory. Inluence o on sintering an integration o supporte Rh particles is stuie uner a wie range o particle sizes an reaction conitions. A brie summary is given in Section THEORETICAL FORMALISM 2.1. Energetics o Supporte Particles. Uner operating conitions, reactants may asorb on supporte tal particles 1761 an aect their morphology an stability, which is closely relate to subsequent Ostwal ripening an integration. It is thereore important to quantiy the energetics o the supporte tal particles in the presence o reactants. To stuy this, we start rom the energetics o a supporte tal particle in the absence o reactants. As shown schematically in Figure 1, a supporte tal particle in a Figure 1. Schematic o supporte tal particle in a spheric segnt with the raius o curvature R an the contact angle α between the particle an support. γ, γ ox, an γ int are the surace energies o the tal particle an support, an the interace energy between tal particle an support, respectively. is the projecte iater o the tal particle on support. spherical segnt can be escribe by the raius o curvature R, contact angle α with respect to the support, expose surace area o the spherical segnt A s =2πR 2 [1 cos (α)], an contact interace area between particle an support A int = πr 2 sin 2 (α). Average energy ΔE (per atom) with respect to ininite size particle (bulk) can be calculate by 1 Δ E = [( + γ + N A s A int ) A int H ah ] (1) where N =4πR 3 α 1 /3Ω is the number o the tal atoms in the particle o interest, an Ω is the molar volu o bulk tal atom, an α 1 =[2 3cos (α) + cos 3 (α)]/4. γ is surace energy o the tal particle, H ah = γ int γ γ ox is ahesion energy between the tal particle an support, γ int is interacial energy between the tal particle an support, an γ ox is surace energy o the support. Base on the Young equation, cos (α)γ = γ ox γ int, one has H ah = [1 + cos (α)]γ. Accoringly, ΔE can be reormulate as 1 Δ E = α γ = Ωγ N A A 3 [ s int cos( )] R Consiering that tal particles may expose ierent acets i with surace energy γ i an corresponing area ratio i over the whole surace area, the overall surace energy γ coul be rewritten as γ = γ i i i The chemical potential (ierentiate energy) Δμ o the supporte tal particle can be erive Δ μ = Δ = Ωγ N ( N E ) 2 R (4) This equation is oten note as the Gibbs Thomson (G T) relation in the literature. 7,11,42,61 Both the average energy an the chemical potential o the particle eine by eq 2 an eq 4 are proportional to the reciprocal o the raius o curvature R an the surace energy γ. Naly, a particle with a smaller raius o curvature R an higher surace energy woul have higher energies an chemical (2) (3) x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

3 Journal o the Arican Chemical Society Figure 2. Energetic iagram o supporte tal particles without (a) an with (b) the presence o reactants. Here, Δμ (R) an Δμ (R) are the chemical potentials o supporte tal particles, E ma an ΔE ma (R) are the ormation energies o monors (the tal aatoms) on support with respect to ininite an inite size tal particle, ΔG is the Gibbs ree energy o asorption o reactants ( in present work) on the tal aatom, an E ma an E carb are the iusion barriers o monors (the tal aatoms an the tal reactant complexes) on support. potential. Depenence o 1/R cos rom the act that the extent o estabilization is proportional to the ratio o surace atoms over the total number o the atoms in the particle. 4,62,63 We note that the MSI, which is essential to the stability o the perse tal particles, is inclue implicitly in these equations. To see this clearly, we consier a mass (volu) conserve particle containing N tal atoms eposite on a certain support. Since a ierent support might have a rather ierent MSI, the contact angle α (an α 1 ) coul change. Base on N =4πR 3 α 1 /3Ω, the raius o curvature R woul change, an so on or the corresponing ΔE an Δμ. Stronger MSI (larger ahesion energy H ah ) woul lea to smaller α (an α 1 ) an larger R, an eventually lower ΔE an Δμ. When α 0, R an thus both ΔE an Δμ approach zero. In this limit, the tal particles woul wet the support orming a twoinsional ilm, an not experience the Gibbs Thomson eect anymore. Corresponing energetics are bulk-like, or even lower when the interaction between the tal an support is stronger than that o the tal tal bon. 49 Uner reaction conitions, reactants may asorb on the tal particles, an corresponing Gibbs ree energy o the asorption woul reuce the surace energy an stabilize the tal particles. Using as an example, the reuction o surace energy Δγ i on the acet i at given temperature T an partial pressure P can be calculate 28 a i i i i Δ γ( T, P) = θ[ E( θ) Δμ ( T, P)]/ A (5) where θ i an A i is the coverage o asorbe an surace unit area o the acet i. E a (θ i ) is the average bining energy o an coverage epenent. The chemical potential o in gas phase is Δμ (T, P) =Δμ (T, P ) +kt ln(p/p ), where k is the Boltzmann constant, an Δμ (T,P ) is the chemical potential o at stanar conition P. 64,65 For reactants at a given T an P, the corresponing coverage θ i can be etermine by E i ( θ) = i [ θi E] θ i a =Δμ ( T, P) where E i (θ i ) is the ierential bining energy o reactants. a i Depenence o E an E on θ i can be obtaine rom the experint or irst-principles theory calculation. The eective surace energy γ o supporte tal particles with asorbates (the symbol with bar represents the variables in the presence o reactants an aopte below without ntion otherwise) becos γ ( T, P) = [ γ + Δγ( T, P)] i i i i By substituting γ in eq 2 an eq 4, average energetics ΔE an chemical potential Δμ o supporte tal particles uner reactants can be obtaine, respectively. Equations 3 an 7 coul be use to construct the equilibrium morphology o the tal particle in the absence an presence o reactants. By minimizing the overall surace energies, the expose acet i an ratio i coul be etermine. It is clear that, in the presence o reactants, the morphology o the tal particle coul change, as ocunte in the literature It is important to note that the particle surace energy an chemical potential now becos a unction o T an P. This will aect the sintering o the supporte tal particles 60 an is inclue in ollowing erivation Ostwal Ripening. As inicate earlier, we ocus in the present work on sintering ominate by surace Ostwal ripening, whereby the tal atoms etach rom small particles with high chemical potential as monors, iuse on the support, an subsequently attach to larger particles with a lower chemical potential. This leas to the growth o larger particles at the expense o smaller particles. The latest erivation o the kinetic equation o Ostwal ripening in the absence o reactants can be oun in the work o Campbell an co-workers. 11 For completeness an consistency with the ollowing erivation in (6) (7) 1762 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

4 Journal o the Arican Chemical Society the presence o reactants, we introuce here the key points with a slightly ierent convention. We start rom the ormation energy o the tal aatoms as monors on the support. Corresponing energetics not only aect the barrier o the etachnt/attachnt o the tal atoms rom/towar the tal particles as cusse below, but also etermine the concentration o monors or, which both are crucial or the sintering rate. The ormation energy ΔE ma o the tal aatoms with respect to a tal particle o raius R (see Figure 2a) is Δ Ema( R) = Ema Δμ ( R) (8) ma E = E E E ma/ox ox B (9) where E ma is the ormation energy o the tal aatoms with respect to ininite size tal (bulk like) particle, E ma/ox is the total energy o the tal aatom on support, E ox is the total energy o the support, an E B is the total energy o the bulk tal. Alternatively, E ma can be calculate rom the cost o the sublimation enthalpy o bulk tal plus subsequent energy gain rom the asorption o an isolate tal atom on the support. 11 The concentration, c s (R), o the tal aatoms in equilibrium with a inite size tal particle o raius R in the ar-iel limit (neglecting the contribution o small vibrational enthalpy 65 ) is thereore written c( R) = s 1 ΔE R exp ma( ) = c 2 a kt 0 eq s Δμ ( R) exp kt (10) where c eq s = exp[ E ma /kt]/a 2 0 is the concentration o the tal aatoms in equilibrium with respect to the ininite size tal particle in the ar-iel limit, a 0 is the lateral lattice constant o support. Uner the steay state or which the ti rate o concentration o the tal aatoms imiately ajacent to a particle is equal to zero, the ti rate equation R/t o the tal particles o raius R coul be erive R XY K E Δ μ * tot ( R ) = exp + exp 2 t X Y R kt kt Δμ ( R) exp kt an E tot is the total activation energy, E = E + E tot ma ma (11) (12) where E ma is the iusion barrier o monors on support, K = ν s Ω/[4πa 2 0 α 1 ], X =2πa 0 R sin(α), Y =2πa 2 0 /ln[l/(r sin (α))], ν s is vibrational requency o the monor on support, L is iusion length require or the monor concentration on support to reach its ar iel limit o c s (R*). R* is the critical particle raius, which is the size o the particle that is in equilibrium with the surrouning aatom concentration an consequently neither grows nor shrinks ue to Ostwal ripening. Equation 11 can be rewritten or two limiting cases, naly, interace control with slow etachnt or attachnt o atoms at the surace o tal particles (Y X) an iusion control (X Y). Depening on interace control or iusion control, the critical raius, R*, woul be 1763 ierent. A rigorous einition or both can be oun in recent work. 66 For a tal particle o raius R less than the critical raius R*, corresponing chemical potential Δμ (R) is higher than that o R*. R/t is negative, the tal atoms leave the tal particles to a to the support, an the tal particle size ecreases. For the tal particles o the raius R larger than R*, R/t becos positive, the tal atoms leave the support to a to the tal particles, an the tal particle size increases. Thus, it is the ierence o Δμ between the tal particles o the raius R an critical raius R* that etermines the growth irection o the iniviual particle o interest an the overall evolution o the size tribution. R* is sensitive to the size an spatial tribution o the tal particles, an woul increase graually with ti. Uner extre cases where the tal particles have ientical size an tribute homogeneously, there will be no ierence o Δμ between any tal particles on the support. This leas to a zero net lux o monors, an Ostwal ripening will be completely suppresse, as inee oun in recent experints. 67 The size or each iniviual tal particle will neither increase nor ecrease, unless the sintering coul procee through the iusion coalescence. On the other han, R/t also epens exponentially on the total activation energy E tot, the sum o the ormation energy o the aatoms E ma an its iusion barrier E ma, which both are etermine by the intrinsic interaction between the tal aatom an support. Since the tal aatoms on the support are oten coorinate unsaturate, E ma is usually highly enothermic an its absolute value is much larger than the chemical potential Δμ o the tal particles. Without consiering the contribution o the iusion barrier E ma, this alreay tells that the absolute ti rate R/t woul be ominate by the total activation energy. For a given tal catalyst, to suppress the Ostwal ripening rate, the optimize support shoul be the one with a higher total activation energy, which coul be achieve by moiying or choosing ierent supports. The kinetics cusse so ar is base on the an-iel approximation, assuming the tal particles are well separate an in equilibrium with ar iel monor. However, its applicability has been subject to much ebate, an the aniel approximation may even break own. 25,46,47,56,68 72 For instance, the long-range equilibrium may not be reache because o the presence o the eects an large iusion barrier, a act o that may lea to graients in monor concentration an thus to local eects. The local eect coul be introuce in aition by the ierence in size an spatial tribution o the tal particles in local vicinity, or a higher tal loaing. In this case, the spatial separation o the tal particles coul approach the iusion length, L, an the particles coul alter the concentration o monors in the vicinity o neighboring particles. These may eviate simulate ecay or growth o the tal particles base on the an-iel approximation rom the asurent. To account or these eects, one shoul employ the so-calle nearest neighbor approach or local correlation approach. 25,47,69 72 In this approach, the sintering o a tal particle o interest will be ecie by its neighbor particles, an a local concentration c s *(R) or critical raius R* varying with spatial an size tribution o neighbor tal particles in the local vicinity shoul be eine an introuce in the above kinetic equations. x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

5 Journal o the Arican Chemical Society 2.3. Reactant Assiste Ostwal Ripening. Uner the reaction conitions, the tal aatoms etache rom the tal particles may be stabilize by reactants orming tal reactant complexes as inicate in Figure 2b, an the concentration o the tal reactant complexes as monors will increase. This woul promote Ostwal ripening o the tal particles an inluence corresponing sintering behavior. To see when an how reactants assist Ostwal ripening an take avantage o the structural simplicity o the mononuclear tal carbonyl complexes, we consier the supporte tal particle uner. The theory can, however, be extene easily to the multinuclear complexes an ierent reactant gases. To stabilize a tal aatom, reactants shoul be able to orm a chemical bon with the tal aatom, an corresponing bining energy E a must be negative. a E = E E n E carb/ox ma/ox (13) where n is the number o coorinate to the tal aatom, E carb/ox is the total energy o the tal carbonyl complexes on support, E ma/ox is the total energy o the tal aatom on support, an E is the total energy o in gas phase. It is evient that the strong interaction between the reactant an the tal aatom is essential or the ormation o the avorable chemical bon. Meanwhile, the local coorination an charge state o the tal aatom is sensitive to the support, which woul inluence the interaction between the tal aatom an the reactant. Hence, given a reactant an tal catalyst o interest, the overall bining strength between the reactant an the tal aatom coul be iate or tune by moiying an choosing ierent supports. For asorption o reactants, loss o gas phase entropy shoul be taken into account. Accoringly, the Gibbs ree energy o asorption ΔG shoul be use a Δ G( T, P) = E n Δμ ( T, P) (14) I the chemical potential o reactants Δμ is too low (low P or high T), the energy gain rom the ormation o the chemical bon cannot compensate the loss o gas-phase entropy. The corresponing asorption woul be enothermic an ΔG is positive. For an exothermic asorption (ΔG < 0), higher Δμ is require an must satisy the ollowing conition Δμ (a) 1 n E a (15) The stabilization o the tal aatoms by asorption o reactants woul lower the ormation energy o the tal aatoms by the amount o ΔG. Corresponing concentration c s (R) o monors in the orm o the tal reactant complexes is Δμ ( R) ΔG c ( R) = c exp exp kt kt s s eq (16) Compare to c s (R) or the tal aatoms in the absence o reactants (eq 10, a unction o Δμ (R) only), c s (R) becos a unction o both Δμ (R) an ΔG (T,P). For reactants interacting strongly with supporte tal particles an the tal aatoms, both coul be stabilize. Since the asorption on supporte tal particles occurs only at the expose surace, the extent o stabilization over the particle by the amount o Δμ Δμ woul be much smaller than that o the iniviual tal aatoms by the amount o ΔG (T,P). As a 1764 result, the concentration o the tal-reactant complexes woul increase exponentially an beco the ominant monors with respect to the tal aatoms when ΔG <0. The increase o concentration o monors woul aect the sintering kinetics. The corresponing ti rate R /t o the supporte tal particles in the presence o reactants via monors in the orm o the tal reactant complexes becos R XY K E Δμ tot ( R ) = exp + exp 2 t X Y R kt kt Δμ ( R) exp kt an E tot is corresponing total activation energy (17) Etot = Ecarb + Ema + ΔG (18) where E carb is the iusion barrier o the tal carboxyl complexes on support. It is clearly seen that the total activation energy becos a unction o the reaction conitions. To see the inluence o reactants on the kinetics, we note that, similar to the kinetics in the absence o reactants, the ierence o the chemical potential Δμ o the tal particles between R* an R etermines the ecay or growth irection, whereas the total activation energy E tot ominates the absolute sintering rate. For reactant-assiste Ostwal ripening, corresponing total activation energy must be lower than that o the tal aatoms as monors in the absence o reactants, naly E < E tot tot (19) Consiering eq 18 an eq 12, this ans carb E +Δ G < E ma (20) For an exothermic asorption (ΔG < 0), the concentration o monors increases exponentially, an the total activation energy woul ecrease by ΔG. However, the sintering rate may not necessarily increase unless eqs 19 an 20 are t. Actually, E tot might be larger than E tot i the corresponing iusion barrier E carb increases to such a value that even counteracts the gain o ΔG, naly, E carb + ΔG > E ma.we note that the reuction o the ormation energy an iusion barrier E carb o the tal reactant complexes coul both lea to a smaller E tot. Their relative values may be very ierent an even reverse compare to that o the tal aatom as monors in the absence o reactants. This may have impact on Ostwal ripening. For instance, or the iusion-controlle Ostwal ripening in the absence o reactants, i the tal reactant complex or in the presence o reactants has lower E carb, the Ostwal ripening coul switch to the interace control, an vice versa. For TiO 2 (110)-supporte Au particles at a iater o 3 nm, Campbell an co-workers estimate that corresponing total activation energy in ultrahigh vacuum was 280 kj/mol, 11 whereas uner oxiation conition, Gooman an co-workers oun that corresponing activation energy was about 10 ± 2 kj/mol only. 25 Though the reason or the ierent activation energies is unclear, the ramatic inluence o the reaction conitions on sintering is evient. This highlights the importance o the in situ stuy on the sintering kinetics, as well ocunte in the literature. 26,46,47,74 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

6 Journal o the Arican Chemical Society 2.4. Reactant Inuce Disintegration. Apart rom reactant-assiste Ostwal ripening via the tal reactant complexes as a transient monor whose ormation shoul be prevente, reactants coul integrate supporte tal particles into the iniviual complexes as the inal prouct spreaing out on support. This is likely when the tal reactant bon energy excees the ierence in energy between the tal tal an tal support bons. 48,49 Since a act o this coul be use to regenerate or reperse the low surace area catalysts, the ormation o the tal reactant complexes shoul be maximize. A rigorous thermoynamic stuy on this topic is escribe here. For reactant inuce integration o a tal particle o the raius R to the iniviual tal reactant complexes, the Gibbs ree energy o asorption o reactants on the tal aatoms shoul be low enough to compensate the cost o the ormation energy o the tal aatom with respect to the tal particle o interest. The easibility coul be escribe by the Gibbs ree energy o integration, ΔG, o the into the tal reactant complexes using as an example again Δ G( R, T, P) =Δ G( T, P) + Ema ΔE ( R) TS (21) where S is the conigurational entropy 75 o the complexes integrate rom the tal particle o interest. To integrate a supporte tal particle, corresponing ΔG must be exothermic (negative). Consiering eq 9, eq 13, an eq 14, the ormula can be reormulate as Δ G( R, T, P) = Ecarb n Δμ ( T, P) ΔE ( R) TS (22) where E carb = E a + E ma is the ormation energy o the tal reactant complexes on support with respect to the ininite size tal particle an reactants in gas phase. It can be oun that the overall value o ΔG is ecie by our parts, naly, the ormation energy o the complexes, the (average) energetics o the tal particles, the chemical potential o reactants in gas phase, an the coniguration entropy ue to the integration. Their inluence an implication are cusse below. First, a lower ormation energy E carb woul lea a lower ΔG. For a given tal catalyst an support, this coul be achieve by varying the composition o reactant gases, which interacts strongly with the tal aatoms. When the tal catalyst an support change, the reactant gas shoul change accoringly or avorable integration, since E carb is etermine by the overall interaction between the reactant, tal, an support. This is inee corroborate by a number o experints. To perse the tal catalysts in experint, the calcination in oxiizing conitions is wiely use: epening on the catalysts an supports, ierent oxiizing reactants ha been aopte. 27,51 55 Secon, ΔG is also sensitive to the reaction conitions an the size o the tal particles. For a given tal particle o the raius R, corresponing chemical potential Δμ () o reactants necessary or integration (ΔG 0) is Δμ () 1 Δ ( n E carb E ( R ) TS ) (23) It is clear that higher chemical potential o reactants (higher P an/or lower T) will be require or the complexes with higher ormation energy an the particle with lower chemical potential. Uner a given reaction conition Δμ, the tal particles o the raius less than R() by consiering eq 2 will be integrate R() 3 Ωγ ( E n Δμ ( T, P) TS) carb 1 (24) It is worth to note that the above criterion or the reactant inuce integration o a particle o interest is rather ierent rom that o the exothermic asorption o reactant on aatoms eine by eq 15. This can be seen rom the ierence o corresponing chemical potential require between eqs 15 an 23 Δμ () Δ μ (a) 1 = Δ ( n E ma E ( R ) TS ) (25) Since the energetics o the tal aatoms on support (E ma )is usually much higher than that o supporte tal particles (ΔE ), the ierence woul be rather positive. In other wors, much higher chemical potential o reactant woul be require or integration. To regenerate the tal catalysts rom the integrate complexes, one shoul lower the chemical potential o reactants (increase T, ecrease P, or apply at sa ti) to ecompose the complexes. By controlling the reucing conitions (T an P) an ti, the tal aatoms release coul start to agglorate an orm cluster until the esire size o the tal particles (thus repersion) is reache. 3. TIO 2 (110) SUPPORTED RH PARTICLES UNDER 3.1. Energetics. The ormalism evelope above was applie to TiO 2 (110) supporte Rh particles uner, because o its importance in hyrogenation an a lot o experintal stuies available or comparison Most energetics require are calculate using vienna ab initio simulation package (VASP) 76 unless ntione otherwise (see etails in Supporting Inormation). Important ata are given in Table 1. Formation energy E ma o Rh aatom on Table 1. Calculate Molar Volu Ω (Å 3 ) o Bulk Rh atom, Formation Energy o Rh Aatom E ma, Bining Energy o on Rh Aatom E a, an Formation Energy o the Rh Carbonyl Complexes E carb or the Monocarbonyl an Dicarbonyl Complexes on TiO 2 (110) a Rh Ω E ma a (mono) 2.18 E E a (i) 4.29 E carb E carb a Energy Unit is ev. (mono) 0.67 (i) 1.44 TiO 2 (110) at the most stable site is highly enothermic by 2.85 ev. This ans a rather low concentration o Rh aatom as monors on TiO 2 (110) an high activation energy o sintering or supporte Rh particles in ultrahigh vacuum. interacts strongly with Rh aatom on top with bining energy E a o 2.18 ev to orm monocarbonyl complexes, as shown in Figure 3. The secon can asorb on Rh aatoms to orm icarbonyl complexes with overall bining energy o 4.29 ev. Corresponing ormation energy E carb o Rh carbonyls with 1765 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

7 Journal o the Arican Chemical Society Table 2. Calculate Surace Energies γ i an Ratio i o Expose Facets o Ininite Size Rh base on Wul Construction a Figure 3. Optimize most stable structures o the Rh carbonyls on TiO 2 (110). The Rh o Rh() sits at the hollow site coorinating with one brige O an one iveol Ti 5. The Rh o Rh() 2 sits at the brige site between the two briging O. Re: O, gray: Ti, cyan: Rh, black: C. respect to bulk Rh are 0.67 an 1.44 ev or the monocarbonyl an the icarbonyl complexes, respectively. The strong bining between an Rh as well as the negative ormation energy or the icarbonyl complexes implies that the presence o woul have signiicant inluence on the sintering an integration, which will be cusse in etail below base on calculate Gibbs ree energy Surace Energy. For the surace energy, γ, o Rh, various orientations incluing (111), (110), (100), (210), (211), (221), (310), an (311) were consiere. Base on the surace energies calculate an Wul construction, the equilibrium morphology o ininite size Rh particle is obtaine an shown in Figure 4a. The expose acets, corresponing acet γ i (ev/å 2 ) i (%) (111) (311) (100) (221) (211) (310) γ Exp a Calculate Average Surace Energy over Wul Construction γ an Experintal Value o Liqui Rh. 77. To see the inluence o asorption on Δγ (eq 5), we consiere asorption on Rh(111). First, the average bining energies, E a, rom coverage o 1/16 ML to 12/16 ML were calculate. In view o experintal results, 78 the asorption was calculate at top site when coverage θ is below 0.25 ML, an at top+hollow sites when θ > 0.25 ML (see Figure S1 in SI). The calculate result is shown in Figure 5. ab Figure 5. Calculate average bining energy E (circle) o on Rh(111) versus coverage. The result is itte to a quaratic polynomial, E ab = θ θ 2, an plotte by a soli line or convenience o the interpolation. The ashe line is the ierential i bining energy, E = θ θ 2. Figure 4. (a) Ininite size Rh morphology rom Wul construction base on the surace energies calculate by irst-principles theory. Optimize cuboctaheral Rh particles base on Wul construction containing 55 (b), 79 (c), an 201 () Rh atoms, an corresponing iaters are 11.5, 13.0, an 17.7 Å. surace energies an ratio, are given in Table 2. It is oun that (111), (311), an (100) acets cover 71%, 9%, an 7% area expose, respectively. The average surace energy over the equilibrium morphology is ev/å 2. The similar surace energy between the average surace an the (111) surace (less than 6 V/Å 2 ) cos rom the small ierence o surace energies between various acets expose, an a higher ratio o (111) surace. The average surace energies calculate agrees well with the asure surace energy (0.125 ev/å 2 ) o liqui Rh. 77 From the average bining energy, the ierential bining i energy E o is obtaine, an plotte in Figure 5 too. The coverage θ o asorbe on Rh(111) versus chemical potential o in gas phase can thereore be calculate by eq 6, an plotte in Figure 6a. Base on these, the reuction o surace energy Δγ (111) on Rh(111) ue to the asorption o is obtaine an plotte in Figure 6b. It can be oun that, when Δμ is lower than 1.52 ev, there is no asorption on Rh(111), an no change in surace energy. With increasing Δμ, starts to asorb an corresponing coverage (111) increases. Accoringly, the reuction o surace energy Δγ increases monotonically. Uner typical experintal conitions o 300 K an 10 1 mbar (Δμ = 0.76 ev), corresponing Δγ (111) is as large as ev/å 2. In comparison to the surace energy o bare Rh(111) γ = ev/å 2, the great inluence o the asorption o reactants on surace energy is clearly seen Chemical Potential. Using the surace energy o ininite-size Rh particle, chemical potential Δμ o the reestaning Rh particles (eq 4) versus the iater =2R was 1766 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

8 Journal o the Arican Chemical Society Figure 6. (a) Calculate coverage θ (ML) on Rh(111) versus chemical potential Δμ o in gas phase. (b) Reuction o surace energy Δγ o Rh(111) ue to asorption, an the itte (111) quaratic polynomial is Δγ = Δμ Δμ or convenience o the interpolation. The relation between Δμ an T at 10 1 mbar an P at T = 300 K is inicate in the top panel. calculate an plotte in Figure 7. The average energy ΔE (eq 2), which is a actor o 1 1 / 2 larger than that o Δμ, is also Figure 7. Chemical potential Δμ (ashe line) an average energy E (soli line) o ree-staning Rh particle in the absence o reactants versus the iater. The surace energy use is the average surace energy o ininite-size Rh base on Wul construction. The open an soli circles are calculate energies o Rh particles (see Figure 4). The zero reerence is ininite-size Rh. plotte or reerence. It can be oun that, with ecrease o rom 100 to 50 Å, Δμ increases slowly up to 0.14 ev with respect to ininite-size Rh. Further ecrease o will lea to a rapi increase o Δμ (), or instance, 0.23 ev at 30 Å an 0.68 ev at 10 Å. It was reporte that chemical potential o supporte tal particles at small size was unerestimate i the sizeinepenent surace energy is use in the G T relation (eq 4). 11,56 The high ratio o the coorinate-unsaturate atoms expose at small size woul increase the surace energy an make it size epenent. To see the size eect on the surace energy an chemical potential o Rh particles consiere here, we constructe three cuboctaheral Rh nanoparticles incluing N = 55, 79, an 201 atoms as plotte in Figure 4b,c, base on above Wul construction o ininite-size Rh. Eective iaters =2R calculate by V = NΩ =4πR 3 /3 are 11.5, 13.0, an 17.7 Å, respectively. The three Rh nanoparticles were ully relaxe. The average energies calculate with respect to ininite size Rh (ΔE ) are 1.09 (N = 55), 0.91 (N = 79), an 0.64 (N = 201) ev/rh atom, an the corresponing chemical potentials Δμ are 0.73, 0.61, an 0.42 ev/rh atom, which all are inclue in Figure 7. For comparison, corresponing chemical potentials rom the G T relation, using the surace energy o ininite-size Rh, are 0.60, 0.53, an 0.39 ev. Although the result base on the G T relation are inee unerestimate, the ierence is only 0.13 ev/rh atom or the smallest Rh nanoparticle ( = 11.5 Å) consiere. For the Rh nanoparticle o = 17.7 Å, the ierence alreay alls to 0.03 ev/rh atom. This inicates that, or Rh particles consiere here, the size eect on the surace energy an chemical potential is moest. To rationalize this result, we note that the optimize surace in-plane lattice constants o Rh particles are oun to ecrease on average by about 0.08 Å (3% lateral contraction), compare to the bulk truncate one. It is likely that, at small particle size, the ecrease o the surace energy rom the smaller in-plane lattice constant compensates the increase o the surace energy rom the higher ratio o the coorinate-unsaturate atoms expose. For Rh particles uner that can asorb, corresponing chemical potential Δμ becos a unction o both the iater an chemical potential Δμ o. For asorption, one shoul in principle consier all possible acets expose. The proceure woul be rather teious, since uner ierent Δμ, asorption coniguration coul be ierent an vary urther on ierent acets. All these woul change corresponing morphology. This may be more involve by consiering the possible size eect upon asorption. As an approximation, we consiere only Rh(111), thus neglecting all other acets expose. This approximation is rationalize by a socalle compensation eect. First, though the ierent acets may interact ierently with, the acets with higher surace energy woul bin more strongly with. Secon, although the smaller particles may interact ierently with, the smaller particle with a higher ratio o coorinate-unsaturate tal atoms woul also bin more strongly with. Actually, as inicate above, the surace energy o Rh(111) is close to the average one rom Wul construction, an the size eect on the chemical potential o ree-staning Rh particles is also small. The valiity o the approximation is justiie inally by the nice agreent o inuce integration o Rh at the broa T, P, an range as cusse below. Beore presenting chemical potential o supporte Rh particles uner or ollowing application, we note that in experint the size o supporte particles is usually asure by the iater o the projection o the particle o the raius R on support, as inicate in Figure 1. To compare with experint, the instea o R is use in the ollowing without ntion otherwise. Consiering the contact angle α, the relation between an R is as ollows: when 0 < α π/2, = 2R sin (α), an when π/2 α π, =2R. For TiO 2 (110) supporte Rh particles, the contact angle α can be estimate rom the experint, 37 where the height/iater ratio o supporte Rh particles was approximately 0.3 at the coverages consiere. Corresponing contact angle α is estimate to be π/3, an R = / x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

9 Journal o the Arican Chemical Society The resulte Δμ versus the reciprocal o an Δμ is plotte in Figure 8. Uner a given Δμ, Δμ increases Figure 8. Contour plot o chemical potential Δμ o TiO 2 (110) supporte Rh particles versus the chemical potential Δμ an the reciprocal o the iater. The contact angle α = π/3 estimate rom the experint 37 was use here an in the ollowing igures without ntion otherwise. linearly with the reciprocal o, as expecte. Whereas uner a given, Δμ ecreases monotonically with increase o Δμ. Naly, supporte Rh particles woul be stabilize graually with increase o chemical potential o (higher P or lower T) ue to the asorption o, which is not surprising at all. Interestingly, larger Rh particles uner lower Δμ coul have the sa chemical potential as those o smaller ones uner higher Δμ, which maniests again the stabilization o reactants. Pronounce inluence o reactants on chemical potential o supporte tal particles woul aect the sintering rate, as oun in recent experints. 60 Its interplay with the stabilization o the tal aatoms woul etermine the overall eect o reactants on sintering an integration Gibbs Free Energy o Disintegration. Base on the above energetics, the inluence o T an P o on sintering an integration o supporte tal particles at given size was stuie irst. In the experint, 37 the iater o Rh particles on TiO 2 (110) as prepare at the submonolayer regi o Θ = 0.01 ML alls in the range o Å, an increase slightly at 0.05 ML. Consiering the limitation o the present theory an experintal error bar at small raius, we set the particles o the iater = 20 Å in calculation. The coniguration entropy S o the ormation o the iniviual complexes ue to the integration o Rh particles at Θ = 0.01 ML is , an shoul be taken into account. Corresponing Gibbs ree energy o integration ΔG (T, P) o the Rh carbonyl complexes was paratrize accoringly Δ G( T, P) = Ecarb n Δμ ( T, P) 3.74 γ ( T, P) T (26) where n = 1 represents or the monocarbonyl Rh() an n = 2 or the icarbonyl Rh() 2. The epenence o ΔG (T,P) on T was stuie at experintal conition o P =10 1 mbar. 37 Calculate ΔG or both Rh() an Rh() 2 are shown in Figure 9 or T in the range K. It can be oun that calculate ΔG ecreases almost linearly with ecrease o T, naly, lower T (higher Δμ ) avors the ormation o the Rh carbonyl complexes. Meanwhile, ormation o Rh() 2 complexes Figure 9. Temperature epenence o the Gibbs ree energy o integration ΔG o the tal reactant complexes o Rh() (re soli line) an Rh() 2 (blue soli line) with respect to TiO 2 (110) supporte Rh particle o the iater = 20 Å uner 10 1 mbar. The vertical ashe lines rom right to let represent the temperature bounary or which Rh() an Rh() 2 complexes becos the ominate monors, an the supporte Rh o interest is integrate into Rh() 2, respectively. becos energetically avorable when T < 750 K because o relatively lower ΔG. Accientally, starts to asorb on Rh particles since the corresponing Δμ > 1.52 ev (Figure 6b). To be exothermic asorption o on Rh aatoms (criterion eine by eq 15), the corresponing Δμ must be higher than 2.18 ev (T 770 K at P =10 1 mbar) or Rh(), an 2.15 ev (T 760 K) or Rh() 2. Consiering ΔG or Rh() an Rh() 2 cross at 750 K, this says that Rh() complexes instea o the tal aatoms will beco the ominant monors or T in the range o [750, 770] K, whereas the ominant complexes will beco Rh() 2 or T in the range o [370, 750] K. Once the iusivity an barrier o the monors (the tal aatoms an the tal reactant complexes) are available, the corresponing total activation energies E tot an E tot can be calculate. Whether Ostwal ripening will be assiste by coul be justiie by eqs 19 an 20. When T 370 K, the corresponing ΔG crosses the zero reerence, criterion eine by eq 23 is satisie. Rh particles o = 20 Å will be integrate to the iniviual Rh() 2, in the case o no kinetics hinrance. Now, we turn to the inluence o P on sintering an integration at experintal conitions o T = 300 K, 37 an the calculate ΔG (T, P) is plotte in Figure 10. It can be oun that ΔG or both Rh() an Rh() 2 ecrease almost linearly with increase o lg(p). Higher P woul avor the ormation o the tal carbonyl complexes. Similar to above, one can in that, or P in the range o [10 25,10 24 ] mbar which coul occur at most experintal conitions, Rh() instea o the tal aatoms will beco the ominant monors, whereas or P in the range o [10 24,10 4 ] mbar, ominant complexes will beco Rh() 2. When P >10 4 mbar, ΔG crosses the zero reerence an becos negative. Accoringly, Rh particles o = 20 Å will be integrate to the iniviual Rh() 2. To see the size epenence o the integration inuce by, T an P are ixe at the experintal conition o 300 K an 10 1 mbar. 37 Uner these conitions (Δμ = 0.76 ev) accoring to Figure 6, the reuction o the surace energy o Rh particles ue to asorption is ev/å 2, an the eective surace energy γ becos ev/å 2, which is much smaller 1768 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,

10 Journal o the Arican Chemical Society Figure 10. Pressure epenence o the Gibbs ree energy o integration ΔG o the tal reactant complexes o Rh() (re soli line) an Rh() 2 (blue soli line) with respect to TiO 2 (110) supporte Rh particle o the iater =20ÅatT = 300 K. The vertical ashe lines rom let to right represent the pressure bounary or which Rh() an Rh() 2 complexes becos the ominate monors, an the supporte Rh o interest is integrate into Rh() 2, respectively. than that o the bare bulk (0.119 ev/å 2 ). On the other han, the ormation o Rh() 2 is thermoynamically more avorable than that o Rh() at this particular conition. Corresponing ΔG or Rh() 2 versus was paratrize an beca only the unction o. Δ G( ) = / (27) The calculate result is plotte in Figure 11. It can be oun that ΔG ecreases with ecrease o an crosses the zero Figure 11. Size epenence o the Gibbs ree energy o integration ΔG o the tal reactant complexes o Rh() 2 with respect to TiO 2 (110) supporte Rh particle uner 10 1 mbar at T = 300 K. reerence at about 60 Å. This tells that Rh particles o the iater less than this woul be integrate into Rh() 2 at T = 300 K an P =10 1 mbar. With urther ecrease o the size, the rate o the integration woul increase ue to the ramatic rop o ΔG proviing an even larger riven orce Discussion. The results above reprouce well the broa T, P, an range over which the sintering an integration occurre in such experints even on ierent supports For TiO 2 (110) supporte Rh particles as prepare at in the range Å, 37 the size ecrease was inee observe at P =10 3 mbar an 300 K. As seen rom Figure 10, P =10 3 mbar alls in the pressure winow o integration (P >10 4 mbar). With graual increase o P up to 10 1 mbar, the experint oun that the rate o the ruption o supporte Rh particles as prepare increases rapily, an appeare completely with ti. This is unerstanable because corresponing pressure alls well in the pressure winow or integration accoring to above calculation. In a ti-resolve in situ Fourier transor inrare asorption spectroscopy (FT-IR) stuy o Al 2 O 3 - supporte Rh particles (298 K an 26.7 kpa), 38 the Rh() complex was oun at the initial exposure o, but only the Rh() 2 complex was observe uner the extene exposure. This can be rationalize by the above calculations which preict that the Rh() 2 complex is thermoynamically more avorable at the experintal conitions. It is likely that the kinetics hinrance o asorbe attaching to the tastable Rh() allows it to be observe by experint. inuce integration o Rh particles supporte on planar SiO 2 was also stuie by Gooman an co-workers using polarization moulation inrare asorption spectroscopy (PM-IRAS). 40 For Rh particles o = 16 Å on average, Rh() 2 PM-IRAS signal was etecte when P >10 5 mbar at 400 K. The increase o signal intensity with P also corroborates well with lower Gibbs ree energy o ormation o the complexes with P plotte in Figure 10. STM experint 37 oun that integrate Rh species rom the Rh particles o =10 20 Å on TiO 2 (110) at 10 1 mbar an 300 K starte to agglorate an orm small Rh particles, when the samples was anneale up to 400 K uner the sa P. Experints oun urther that, when annealing T was increase urther to 600 K, the average iater o the Rh particles attaine a value o 55 Å. In contrast, without the pretreatnt o, the Rh particles as prepare in the range Å attaine only the average iater o 35 Å when the samples were anneale at 900 K uner ultrahigh vacuum. As shown in Figure 9, when T is higher than 370 K at 10 1 mbar, ΔG becos positive, an alls in the temperature winow ominate by Rh() 2 monors. Since the concentration o the corresponing monors is much higher than that o the tal aatoms in the absence o, aggloration to the larger tal particles woul be promote. In terms o the size eect, experints 37 oun that uner 10 1 mbar an at 300 K, the Rh particles o =10 20 Å supporte on TiO 2 (110) were rapily integrate into atomically perse species, while the process was slow or those with =30 40 Å, an i not occur or those with = Å particles, even at higher P an extene exposure ti. These experintal results are corroborate again by our result shown in Figure 11, which inicates that the Rh particles o the iater less than 60 Å woul be integrate by, whereas the larger one woul be resistant to the integration. Similar size epenence o the integration by was also oun on SiO 2 supporte Rh particles. 40 In that work, the PM- IRAS intensity o the Rh() 2 complexes integrate rom the Rh particles o = 16 Å was oun at 10 1 mbar an 400 K, but no Rh() 2 signal was etecte or the Rh particles o = 37 Å uner sa conitions. Excellent agreent between theory an experint over the broa range o temperature, pressure, an particle size justiies the theory evelope an approximation o surace energy o supporte Rh particles. This also shows that the ormation o the tal reactant complexes as avorable monors plays a crucial role in the sintering an integration o supporte tal particles uner reaction conitions. Depening on the conitions, it coul act not only as transient monors to assist 1769 x.oi.org/ /ja J. Am. Chem. Soc. 2013, 135,